A strong firewall should help you keep an eye on attempts to contact your computer. It should also alert you if an application on your machine is attempting to contact another computer. Keep a close eye on these messages. Some are harmless or even beneficial -- you wouldn't want to block your anti-virus program from downloading the latest updates. But others can be signs that someone is trying to access your information or control your machine from a remote location.
Most Web browsers have security settings that can help you keep your network safe. Several will warn you if you are about to visit a site known to host malware. You can also adjust settings such as whether or not your browser will accept cookies or run Java applications. Disabling cookies, Java and other options will help keep your network safe but it will also affect your browsing experience. You may not be able to interact with sites the way the webmaster intended if you turn off these options.
Administrators of commercial computer networks sometimes rely on special software and hardware called intrusion detection systems (IDS). These systems monitor data traffic across host computers and networks. A good IDS can compare this data against known malware patterns and alert the administrator if there's a problem. But that's a solution for a much larger computer network than your typical home network.
Most anti-virus software won't detect an intruder. But you might discover a malware application that makes intrusions possible. Run anti-virus software frequently to make sure your system is safe. You should also be sure to install updates and patches for your operating system as they become available. These patches may help seal up vulnerabilities on your computer.